CARACAS, March 24 (Reuters) - A pregnant woman was shot dead near Caracas and a soldier was killed in the western state of Merida, officials said, as the death toll from weeks of anti-government protests in Venezuela rose to 36 on Monday.

Supporters of both sides and members of the security forces have been among those killed in the nation's worst unrest in a decade, sparked by demonstrations against socialist President Nicolas Maduro that began last month.

Francisco Garces, mayor of Guaicaipuro municipality near the capital and a member of the ruling Socialist Party, said the 28-year-old pregnant woman was shot dead on Sunday during a protest.

The state prosecutor's office said the woman, identified as Adriana Urquiola, was shot after getting off a public bus halted by a makeshift barricade rigged up by protesters.

In the western state of Merida, a National Guard sergeant died on Monday after being shot in the neck during clashes there, the military said.

General Padrino Lopez, head of the armed forces' strategic operational command, said the sergeant, Miguel Parra, and Urquiola were "assassinated at the barricades of terror."

The street barriers are flashpoints for violence between radical supporters of both sides, who are sometimes armed. Members of the security forces have also come under fire from nearby buildings as they try to dismantle the barricades.

Merida and neighboring Tachira state, on the border with Colombia, have been hardest hit by the unrest.

Last week, intelligence agents arrested the opposition mayor of San Cristobal city in Tachira during a visit to Caracas and accused him of fomenting "civil rebellion."

Another mayor of an opposition-run municipality in central Carabobo state was jailed for 10 months after a court ruled he failed to comply with an order to take down barricades.

On Monday, Maduro said the opposition's two-time presidential candidate Henrique Capriles must heed such rulings in his role as governor of Miranda state, which includes parts of the capital, or give up the position.

"I think the Supreme Court has a great responsibility to make this governor comply with the law and with the citizens," Maduro said in a telephone call to state TV. "If not, he goes. The governor of Miranda will definitely go."

MOVES AGAINST LAWMAKER

The protests began in February with sporadic demonstrations by university students. They intensified after three people were killed following a Feb. 12 rally in downtown Caracas.

The demonstrators are demanding political change and an end to high inflation, shortages of basic foods, and one of the highest rates of violent crime in the world.

Alongside the moves against the opposition mayors, Maduro's supporters in Congress have requested a criminal investigation of an opposition legislator for crimes including treason relating to the protests.

The National Assembly president, Diosdado Cabello, said on Monday that Maria Corina Machado was no longer a lawmaker after she spoke out against the government last week during a meeting of the Organization of American States (OAS).

"She's no longer a deputy," Cabello told reporters, adding that her acceptance of an offer from Panama to speak at the OAS was unconstitutional. "We're giving instructions that this woman not be allowed back into parliament for this session."

A constitutional lawyer consulted last week by Reuters said Machado could not lose her parliamentary immunity without approval from both the state prosecutor and the Supreme Court, and then another vote in the National Assembly.

Machado, a 46-year-old engineer elected to Congress in 2010, is frequently pilloried by Socialist Party supporters as an out-of-touch elitist with an especially wealthy background.

Despite her high profile during the protests and hero status for hardline opponents, others in the opposition's more moderate ranks criticize her as shrill and overly confrontational.

The lawmaker, who says the president and his allies preside over a dictatorship, said she would not give up without a fight.

"Mr. Cabello, I will remain a deputy for as long as the people of Venezuela want me to," Machado said on Twitter.

The demonstrators are demanding the president resign, while Maduro says they want a coup like the one 12 years ago that briefly ousted his predecessor, the late Hugo Chavez.

The numbers of protesters are far fewer than those who turned out against Chavez in 2002, and there have been no signs that the current unrest threatens to topple Maduro. (Additional reporting by Eyanir Chinea in Caracas and Javier Faria in San Cristobal; Editing by Andrew Cawthorne, Andrew Hay, Paul Simao and Lisa Shumaker)